How does one prove this, independently of charts used for a manifold? I'm not sure where to even begin. I've seen people do it with equivalence classes of curves defining the TM but I can't visualize how to do that and then I have a problem with the surjectivity requirement. Sorry if this is vague, if you need more information please let me know.
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Nevermind, I figured it out using https://math.stackexchange.com/questions/3486640/does-the-linear-isomorphism-between-the-tangent-space-at-a-point-and-mathbbr and https://math.stackexchange.com/questions/3107806/identification-of-the-tangent-space-of-a-manifold-and-the-tangent-vectors-to-cur – James Shelton Mar 02 '21 at 04:45